{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# 1D Heat Equation"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Heat transfer in 1D is governed by the following PDE (See [here](https://ocw.mit.edu/courses/mathematics/18-303-linear-partial-differential-equations-fall-2006/lecture-notes/heateqni.pdf) for more info):"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{\\partial u}{\\partial t} = \\kappa \\frac{\\partial^2 u}{\\partial x^2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\kappa = K_0/c\\rho$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{x}=x/L_*$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{t}=t/T_*$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\hat{u}(\\hat{x},\\hat{t})=u(x,t)/U_*$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{\\partial \\hat{u}}{\\partial \\hat{t}} = \\frac{T_* \\kappa}{L^2_*} \\frac{\\partial^2 \\hat{u}}{\\partial \\hat{x}^2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$T_* = L^2_*/ \\kappa$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$\\frac{\\partial \\hat{u}}{\\partial \\hat{t}} = \\frac{\\partial^2 \\hat{u}}{\\partial \\hat{x}^2}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Suppose the initial temperature distibution in a 1D rod is constant i.e. $f(x) = u_0$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$u(x,t) = \\sum ^\\inf _{n=1} B_n sin(n\\pi x)e^{-n^2 \\pi^2 t}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "where"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$B_n = 2u_0 \\int_0^1 sin(n \\pi x) dx$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$u(x,t) = \\frac{4u_0}{\\pi} \\sum_{n=1}^{\\inf} \\frac{sin((2n-1)\\pi x)}{(2n-1)} e^{-(2n-1)^2\\pi^2 t}$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$u(x,t) = \\frac{4u_0}{\\pi}\\left( sin(\\pi x)e^{-\\pi^2 t} + \\frac{sin(3\\pi x)}{3}e^{-9\\pi ^2 t} + ... \\right)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:06.676516Z",
     "iopub.status.busy": "2021-06-24T11:30:06.674185Z",
     "iopub.status.idle": "2021-06-24T11:30:07.003518Z",
     "shell.execute_reply": "2021-06-24T11:30:07.002061Z"
    }
   },
   "outputs": [],
   "source": [
    "import numpy as np\n",
    "np.random.seed(10)\n",
    "np.set_printoptions(5)\n",
    "import matplotlib.pyplot as plt"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:07.011304Z",
     "iopub.status.busy": "2021-06-24T11:30:07.009707Z",
     "iopub.status.idle": "2021-06-24T11:30:07.012842Z",
     "shell.execute_reply": "2021-06-24T11:30:07.014103Z"
    }
   },
   "outputs": [],
   "source": [
    "def approx_sol(u0, x, t, N):\n",
    "    out = 0\n",
    "    for i in range(N):\n",
    "        n = i+1\n",
    "        out += np.sin((2*n-1)*np.pi*x) * np.exp(-(2*n-1)**2 * np.pi**2 * t) / (2*n-1)\n",
    "    return (4*u0/np.pi)*out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:07.138288Z",
     "iopub.status.busy": "2021-06-24T11:30:07.102003Z",
     "iopub.status.idle": "2021-06-24T11:30:07.284589Z",
     "shell.execute_reply": "2021-06-24T11:30:07.283587Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Solution maximum: 0.46835\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "u0 = 1.0\n",
    "x = np.linspace(0, 1, 101)\n",
    "plt.figure()\n",
    "for t in [1/np.pi**2]:\n",
    "    sol = approx_sol(u0, x, t, N=10000)\n",
    "    plt.plot(x, sol)\n",
    "    print(f\"Solution maximum: {sol.max():.5f}\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:07.368585Z",
     "iopub.status.busy": "2021-06-24T11:30:07.332516Z",
     "iopub.status.idle": "2021-06-24T11:30:07.754952Z",
     "shell.execute_reply": "2021-06-24T11:30:07.756107Z"
    }
   },
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "t = np.linspace(0, 1, 101)\n",
    "plt.figure()\n",
    "for x in [0.01, 0.25, 0.5, 0.75]:\n",
    "    plt.plot(t, approx_sol(u0, x, t, N=10000))\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:07.763187Z",
     "iopub.status.busy": "2021-06-24T11:30:07.761751Z",
     "iopub.status.idle": "2021-06-24T11:30:07.767607Z",
     "shell.execute_reply": "2021-06-24T11:30:07.766419Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Thermal diffusivity (kappa): 1.15955e-04\n"
     ]
    }
   ],
   "source": [
    "# Copper\n",
    "K0 = 400\n",
    "rho = 8960\n",
    "Cp = 385\n",
    "kappa = K0/(rho*Cp)\n",
    "print(f\"Thermal diffusivity (kappa): {kappa:.5e}\")\n",
    "l = 1.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:07.776832Z",
     "iopub.status.busy": "2021-06-24T11:30:07.775452Z",
     "iopub.status.idle": "2021-06-24T11:30:07.778315Z",
     "shell.execute_reply": "2021-06-24T11:30:07.779448Z"
    }
   },
   "outputs": [],
   "source": [
    "def approx_sol_dim(u0, x, t, N, kappa, l):\n",
    "    out = 0\n",
    "    for i in range(N):\n",
    "        n = i+1\n",
    "        out += np.sin((2*n-1)*np.pi*x/l) * np.exp(-(2*n-1)**2 * np.pi**2 * t * kappa/l**2) / (2*n-1)\n",
    "    return (4*u0/np.pi)*out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:07.861876Z",
     "iopub.status.busy": "2021-06-24T11:30:07.825525Z",
     "iopub.status.idle": "2021-06-24T11:30:08.296374Z",
     "shell.execute_reply": "2021-06-24T11:30:08.297523Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Diffusion time: 14.56323 mins\n",
      "Solution maximum: 0.76754\n",
      "Solution maximum: 0.46835\n",
      "Solution maximum: 0.28410\n"
     ]
    },
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(0, l, 101)\n",
    "plt.figure()\n",
    "t_diffusion = (1/np.pi**2)/(kappa/l**2)\n",
    "print(f\"Diffusion time: {t_diffusion/60:.5f} mins\")\n",
    "for t in np.linspace(0.5, 1.5, 3)*t_diffusion:\n",
    "    sol = approx_sol_dim(u0, x, t, 10000, kappa, l)\n",
    "    plt.plot(x, sol)\n",
    "    print(f\"Solution maximum: {sol.max():.5f}\")\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.305529Z",
     "iopub.status.busy": "2021-06-24T11:30:08.304418Z",
     "iopub.status.idle": "2021-06-24T11:30:08.752579Z",
     "shell.execute_reply": "2021-06-24T11:30:08.751347Z"
    }
   },
   "outputs": [],
   "source": [
    "import openpnm as op\n",
    "%config InlineBackend.figure_formats = ['svg']\n",
    "ws = op.Workspace()\n",
    "spacing = 1e-2\n",
    "net = op.network.Cubic(shape=[101, 1, 1], spacing=spacing)\n",
    "# translate to origin\n",
    "net['pore.coords'] -= np.array([spacing, spacing, spacing])/2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.759149Z",
     "iopub.status.busy": "2021-06-24T11:30:08.757725Z",
     "iopub.status.idle": "2021-06-24T11:30:08.763663Z",
     "shell.execute_reply": "2021-06-24T11:30:08.762417Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Length: 1.00000\n"
     ]
    }
   ],
   "source": [
    "l = net['pore.coords'][:, 0].max() - net['pore.coords'][:, 0].min()\n",
    "print(f\"Length: {l:.5f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.770879Z",
     "iopub.status.busy": "2021-06-24T11:30:08.769765Z",
     "iopub.status.idle": "2021-06-24T11:30:08.772110Z",
     "shell.execute_reply": "2021-06-24T11:30:08.772994Z"
    }
   },
   "outputs": [],
   "source": [
    "geo = op.geometry.GenericGeometry(network=net, pores=net.Ps, throats=net.Ts)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.782641Z",
     "iopub.status.busy": "2021-06-24T11:30:08.781517Z",
     "iopub.status.idle": "2021-06-24T11:30:08.784636Z",
     "shell.execute_reply": "2021-06-24T11:30:08.785773Z"
    }
   },
   "outputs": [],
   "source": [
    "geo['pore.diameter'] = spacing\n",
    "geo['throat.diameter'] = spacing\n",
    "geo['throat.length'] = spacing\n",
    "geo['throat.area'] = spacing**2\n",
    "geo['pore.area'] = spacing**2\n",
    "geo['pore.volume'] = spacing**3\n",
    "geo['throat.volume'] = 0.0"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.799840Z",
     "iopub.status.busy": "2021-06-24T11:30:08.798370Z",
     "iopub.status.idle": "2021-06-24T11:30:08.801342Z",
     "shell.execute_reply": "2021-06-24T11:30:08.802442Z"
    }
   },
   "outputs": [],
   "source": [
    "phase = op.phases.GenericPhase(network=net)\n",
    "phase['pore.conductivity'] = kappa\n",
    "phys = op.physics.GenericPhysics(network=net, geometry=geo, phase=phase)\n",
    "c = 1.0  # mol/m^3\n",
    "phys['throat.conductance'] = c*kappa*geo['throat.area']/geo['throat.length']"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.814011Z",
     "iopub.status.busy": "2021-06-24T11:30:08.812616Z",
     "iopub.status.idle": "2021-06-24T11:30:08.951817Z",
     "shell.execute_reply": "2021-06-24T11:30:08.950856Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Effective conductivity: 1.15955e-04, kappa: 1.15955e-04, do they match? True\n"
     ]
    }
   ],
   "source": [
    "alg = op.algorithms.FourierConduction(network=net, phase=phase)\n",
    "alg.settings['conductance'] = 'throat.conductance'\n",
    "alg.set_value_BC(pores=[0], values=1.0)\n",
    "alg.set_value_BC(pores=[-1], values=0.0)\n",
    "alg.run()\n",
    "K_eff = alg.calc_effective_conductivity(domain_length=l, domain_area=spacing**2)[0]\n",
    "print(f\"Effective conductivity: {K_eff:.5e}, kappa: {kappa:.5e}, do they match? {np.allclose(K_eff, kappa)}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:08.964984Z",
     "iopub.status.busy": "2021-06-24T11:30:08.961470Z",
     "iopub.status.idle": "2021-06-24T11:30:09.168078Z",
     "shell.execute_reply": "2021-06-24T11:30:09.168555Z"
    }
   },
   "outputs": [
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      "text/plain": [
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     },
     "metadata": {
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     "output_type": "display_data"
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   ],
   "source": [
    "alg = op.algorithms.TransientReactiveTransport(network=net, phase=phase)\n",
    "alg.settings.update({'conductance': 'throat.conductance', \n",
    "                     'quantity': 'pore.temperature',\n",
    "                     't_initial': 0, \n",
    "                     't_final': 880, \n",
    "                     't_step': 87, \n",
    "                     't_output': 10,\n",
    "                     't_tolerance': 1e-12, \n",
    "                     't_precision': 12, \n",
    "                     'rxn_tolerance': 1e-12, \n",
    "                     't_scheme': 'implicit'})\n",
    "alg.set_IC(values=u0)\n",
    "alg.set_value_BC(pores=[0], values=0.0)\n",
    "alg.set_value_BC(pores=[-1], values=0.0)\n",
    "alg.run()\n",
    "res = alg.results()\n",
    "times = list(res.keys())\n",
    "times.sort()\n",
    "plt.figure()\n",
    "for time in times:\n",
    "    plt.plot(alg[time])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:09.172110Z",
     "iopub.status.busy": "2021-06-24T11:30:09.171552Z",
     "iopub.status.idle": "2021-06-24T11:30:09.174291Z",
     "shell.execute_reply": "2021-06-24T11:30:09.173707Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Diffusion time: 873.79389 sec\n"
     ]
    }
   ],
   "source": [
    "print(f\"Diffusion time: {t_diffusion:.5f} sec\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:09.177982Z",
     "iopub.status.busy": "2021-06-24T11:30:09.177431Z",
     "iopub.status.idle": "2021-06-24T11:30:09.179842Z",
     "shell.execute_reply": "2021-06-24T11:30:09.180222Z"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "True"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "'pore.temperature@870' in times"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:09.257694Z",
     "iopub.status.busy": "2021-06-24T11:30:09.221228Z",
     "iopub.status.idle": "2021-06-24T11:30:09.406584Z",
     "shell.execute_reply": "2021-06-24T11:30:09.405368Z"
    }
   },
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
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     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    },
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum error: 0.02174\n"
     ]
    }
   ],
   "source": [
    "plt.figure()\n",
    "x = net['pore.coords'][:, 0]\n",
    "a = alg['pore.temperature@870']\n",
    "b = approx_sol_dim(u0, x, 870.0, 10000, kappa, l)\n",
    "plt.plot(a)\n",
    "plt.plot(b, '--')\n",
    "plt.show()\n",
    "print(f\"Maximum error: {np.max(a-b):.5f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:09.413876Z",
     "iopub.status.busy": "2021-06-24T11:30:09.412776Z",
     "iopub.status.idle": "2021-06-24T11:30:09.415068Z",
     "shell.execute_reply": "2021-06-24T11:30:09.415955Z"
    }
   },
   "outputs": [],
   "source": [
    "def approx_sol_inhom(u1, x, t, N):\n",
    "    out = 0\n",
    "    for i in range(N):\n",
    "        n = i+1\n",
    "        out += ((-1)**n)/n * np.sin(n*np.pi*x) * np.exp(-n**2 * np.pi**2 * t)\n",
    "    return u1*x + (2*u1/np.pi)*out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "metadata": {
    "execution": {
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     "iopub.status.busy": "2021-06-24T11:30:09.459350Z",
     "iopub.status.idle": "2021-06-24T11:30:10.174078Z",
     "shell.execute_reply": "2021-06-24T11:30:10.175238Z"
    }
   },
   "outputs": [
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   "source": [
    "u1 = 1.0\n",
    "x = np.linspace(0, 1, 101)\n",
    "plt.figure()\n",
    "for t in np.array([0.01, 0.1, 0.5, 1.0, 1.5, 3])/np.pi**2:\n",
    "    sol = approx_sol_inhom(u1, x, t, N=10000)\n",
    "    plt.plot(x, sol)\n",
    "plt.show()"
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  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:10.182446Z",
     "iopub.status.busy": "2021-06-24T11:30:10.181071Z",
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     "shell.execute_reply": "2021-06-24T11:30:10.184806Z"
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   },
   "outputs": [],
   "source": [
    "def approx_sol_inhom_dim(u1, x, t, N, kappa, l):\n",
    "    out = 0\n",
    "    for i in range(N):\n",
    "        n = i+1\n",
    "        out += ((-1)**n)/n * np.sin(n*np.pi*x/l) * np.exp(-n**2 * np.pi**2 * t * kappa/l**2)\n",
    "    return u1*x/l + (2*u1/np.pi)*out"
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     "shell.execute_reply": "2021-06-24T11:30:10.355519Z"
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      "text/plain": [
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     },
     "metadata": {
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     "output_type": "display_data"
    }
   ],
   "source": [
    "alg = op.algorithms.TransientReactiveTransport(network=net, phase=phase)\n",
    "alg.settings.update({'conductance': 'throat.conductance', \n",
    "                     'quantity': 'pore.temperature',\n",
    "                     't_initial': 0, \n",
    "                     't_final': 880, \n",
    "                     't_step': 87, \n",
    "                     't_output': 10,\n",
    "                     't_tolerance': 1e-12, \n",
    "                     't_precision': 12, \n",
    "                     'rxn_tolerance': 1e-12, \n",
    "                     't_scheme': 'implicit'})\n",
    "alg.set_IC(values=0.0)\n",
    "alg.set_value_BC(pores=[0], values=0.0)\n",
    "alg.set_value_BC(pores=[-1], values=u1)\n",
    "alg.run()\n",
    "res = alg.results()\n",
    "times = list(res.keys())\n",
    "times.sort()\n",
    "plt.figure()\n",
    "for time in times:\n",
    "    plt.plot(alg[time])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 22,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:10.434090Z",
     "iopub.status.busy": "2021-06-24T11:30:10.397702Z",
     "iopub.status.idle": "2021-06-24T11:30:10.661528Z",
     "shell.execute_reply": "2021-06-24T11:30:10.660306Z"
    }
   },
   "outputs": [
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      "0.0\n"
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   ],
   "source": [
    "plt.figure()\n",
    "x = net['pore.coords'][:, 0]\n",
    "a = alg['pore.temperature@870']\n",
    "b = approx_sol_inhom_dim(u0, x, 870.0, 10000, kappa, l)\n",
    "plt.plot(a)\n",
    "plt.plot(b, '--')\n",
    "plt.show()\n",
    "print(np.max(a-b))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Now consider a source term $sin(\\pi x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$u_{t} = u_{xx} + sin(\\pi x)$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "subject to B.Cs"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "$u(0) = u(1) = 0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:10.667785Z",
     "iopub.status.busy": "2021-06-24T11:30:10.666772Z",
     "iopub.status.idle": "2021-06-24T11:30:10.668852Z",
     "shell.execute_reply": "2021-06-24T11:30:10.669659Z"
    }
   },
   "outputs": [],
   "source": [
    "def approx_sol_source(x, t):\n",
    "    return (np.sin(np.pi*x)/(np.pi**2))*(1-np.exp(-np.pi**2 * t))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:10.675344Z",
     "iopub.status.busy": "2021-06-24T11:30:10.674326Z",
     "iopub.status.idle": "2021-06-24T11:30:10.676318Z",
     "shell.execute_reply": "2021-06-24T11:30:10.677054Z"
    }
   },
   "outputs": [],
   "source": [
    "def approx_sol_source_dim(x, t, kappa, l):\n",
    "    return (np.sin(np.pi*x/l)/(np.pi**2))*(1-np.exp(-np.pi**2 * t * kappa/l**2))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
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     "iopub.status.idle": "2021-06-24T11:30:10.841951Z",
     "shell.execute_reply": "2021-06-24T11:30:10.843120Z"
    }
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   "source": [
    "x = np.linspace(0, 1, 101)\n",
    "plt.figure()\n",
    "for t in np.array([0.01, 0.1, 0.5, 1.0, 1.5, 3, 100])/np.pi**2:\n",
    "    sol = approx_sol_source(x, t)\n",
    "    plt.plot(x, sol)\n",
    "plt.show()"
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  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:10.862951Z",
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     "iopub.status.idle": "2021-06-24T11:30:12.116829Z",
     "shell.execute_reply": "2021-06-24T11:30:12.117536Z"
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   "outputs": [
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "Qf = 2000*net['pore.volume']/(Cp*rho)\n",
    "Q = Qf*np.sin(np.pi*x)\n",
    "phys['pore.source.S1'] = 0.0\n",
    "phys['pore.source.S2'] = Q\n",
    "phys['pore.source.rate'] = Q\n",
    "\n",
    "alg = op.algorithms.TransientReactiveTransport(network=net, phase=phase)\n",
    "alg.settings.update({'conductance': 'throat.conductance', \n",
    "                     'quantity': 'pore.temperature',\n",
    "                     't_initial': 0, \n",
    "                     't_final': 10000, \n",
    "                     't_step': 100, \n",
    "                     't_output': 100,\n",
    "                     't_tolerance': 1e-7, \n",
    "                     't_precision': 12, \n",
    "                     'rxn_tolerance': 1e-7, \n",
    "                     't_scheme': 'implicit'})\n",
    "alg.set_IC(values=0.0)\n",
    "alg.set_value_BC(pores=[0], values=0.0)\n",
    "alg.set_value_BC(pores=[-1], values=0.0)\n",
    "alg.set_source(propname='pore.source', pores=net.pores()[1:-1])\n",
    "alg.run()\n",
    "res = alg.results()\n",
    "times = list(res.keys())\n",
    "times.sort()\n",
    "plt.figure()\n",
    "for time in times:\n",
    "    plt.plot(alg[time])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:12.123401Z",
     "iopub.status.busy": "2021-06-24T11:30:12.122531Z",
     "iopub.status.idle": "2021-06-24T11:30:12.126429Z",
     "shell.execute_reply": "2021-06-24T11:30:12.127002Z"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Maximum temperature: 0.50278\n"
     ]
    }
   ],
   "source": [
    "print(f\"Maximum temperature: {alg['pore.temperature@4500'].max():.5f}\")"
   ]
  },
  {
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     "output_type": "display_data"
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    {
     "name": "stdout",
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     "text": [
      "Maximum error: 0.40205\n"
     ]
    }
   ],
   "source": [
    "plt.figure()\n",
    "x = net['pore.coords'][:, 0]\n",
    "a = alg['pore.temperature@4500']\n",
    "b = approx_sol_source_dim(x, 4500.0, kappa, l)\n",
    "plt.plot(a)\n",
    "plt.plot(b, '--')\n",
    "plt.show()\n",
    "print(f\"Maximum error: {np.max(a-b):.5f}\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:12.241287Z",
     "iopub.status.busy": "2021-06-24T11:30:12.240182Z",
     "iopub.status.idle": "2021-06-24T11:30:12.242451Z",
     "shell.execute_reply": "2021-06-24T11:30:12.243344Z"
    }
   },
   "outputs": [],
   "source": [
    "def crazy_source(x, t, N):\n",
    "    # https://www.math.upenn.edu/~deturck/m241/inhomogeneous.pdf\n",
    "    out = 0\n",
    "    for n in range(N):\n",
    "        a = 4*((2*n+1)**2 * np.pi**2 - 2)*np.exp(-(2*n+1)**2 * np.pi**2 * t)/((2*n+1)**3 * np.pi**3 * ((2*n+1)**2 *np.pi**2 - 1))\n",
    "        b = 4*np.exp(-t)/((2*n+1)*np.pi*((2*n+1)**2*np.pi**2-1))\n",
    "        c = np.sin((2*n+1)*np.pi*x)\n",
    "        out += (a+b)*c\n",
    "    return out"
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  {
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   "execution_count": 30,
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     "iopub.status.idle": "2021-06-24T11:30:13.183263Z",
     "shell.execute_reply": "2021-06-24T11:30:13.184396Z"
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   "source": [
    "x = np.linspace(0, 1, 101)\n",
    "plt.figure()\n",
    "times = np.array([0.01, 0.1, 0.5, 1.0, 1.5, 3])/np.pi**2\n",
    "for t in times:\n",
    "    sol = crazy_source(x, t, 10000)\n",
    "    plt.plot(x, sol)\n",
    "plt.show()"
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  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:13.205373Z",
     "iopub.status.busy": "2021-06-24T11:30:13.203971Z",
     "iopub.status.idle": "2021-06-24T11:30:13.273562Z",
     "shell.execute_reply": "2021-06-24T11:30:13.274686Z"
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   "source": [
    "plt.figure()\n",
    "x = np.linspace(0, 2, 101)\n",
    "plt.plot(1-np.abs(x-1))\n",
    "plt.show()"
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  },
  {
   "cell_type": "code",
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     "shell.execute_reply": "2021-06-24T11:30:13.368671Z"
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      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
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     "output_type": "display_data"
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   ],
   "source": [
    "plt.figure()\n",
    "plt.plot(2*x-x**2)\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 33,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:13.377685Z",
     "iopub.status.busy": "2021-06-24T11:30:13.376516Z",
     "iopub.status.idle": "2021-06-24T11:30:13.378884Z",
     "shell.execute_reply": "2021-06-24T11:30:13.379803Z"
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   "outputs": [],
   "source": [
    "def qn(n, x):\n",
    "    return (8/(n**2 * np.pi**2))*np.sin(n*np.pi*x/2)\n",
    "def an(n, x):\n",
    "    return (4/(n**2 * np.pi**2))*qn(n, x)\n",
    "def cn(n):\n",
    "    return (16/(n**3 * np.pi**3))*(1-np.cos(n*np.pi))\n",
    "def bn(n, x):\n",
    "    return cn(n) - an(n, x)\n",
    "def approx_sol_sawtooth(x, t, N):\n",
    "    #https://faculty.uca.edu/darrigo/Students/M4315/Fall%202005/sep-var.pdf\n",
    "    out = 0\n",
    "    for i in range(N):\n",
    "        n = i+1\n",
    "        np2 = (n*np.pi/2)\n",
    "        e = np.exp(-((np2**2) * t))\n",
    "        out += ( an(n, x) + bn(n, x) * e ) * np.sin(np2*x)\n",
    "    \n",
    "    return out"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 34,
   "metadata": {
    "execution": {
     "iopub.execute_input": "2021-06-24T11:30:13.456052Z",
     "iopub.status.busy": "2021-06-24T11:30:13.419735Z",
     "iopub.status.idle": "2021-06-24T11:30:16.247558Z",
     "shell.execute_reply": "2021-06-24T11:30:16.248719Z"
    }
   },
   "outputs": [
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    "for t in [0.0, 0.01, 0.1, 0.5, 1.0, 2.0, 3.0, 100.0]:\n",
    "    sol = approx_sol_sawtooth(x, t, 10000)\n",
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    "plt.show()"
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